1. FOURIER ANALYSIS PART 2: Technicalities, FFT & system analysis
Maria Elena Angoletta
AB/BDI
DISP 2003, 27 February 2003

2. TOPICS 2. DFT resolution - improvement
1. DFT windows
2. DFT resolution - improvement
3. Efficient DFT calculation: FFT
4. Hints on system spectral analysis
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3. DFT – Window characteristics
Finite discrete sequence  spectrum convoluted with rectangular window spectrum.
Leakage amount depends on chosen window & on how signal fits into the window.
Resolution: capability to distinguish different tones. Inversely proportional to main-lobe width. Wish: as high as possible.
(1)
Peak-sidelobe level: maximum response outside the main lobe. Determines if small signals are hidden by nearby stronger ones.
Wish: as low as possible.
(2)
Sidelobe roll-off: sidelobe decay per decade. Trade-off with (2).
(3)
Rectangular window
Several windows used (application-dependent): Hamming, Hanning, Blackman, Kaiser ...
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4. In time it reduces end-points discontinuities.
DFT of main windows
Some window functions
Windowing reduces leakage by minimising sidelobes magnitude.
Sampled sequence
In time it reduces end-points discontinuities.
Non windowed
Windowed
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5. Common windows characteristics
DFT - Window choice
Common windows characteristics
Far & strong interfering components  high roll-off rate.
Near & strong interfering components  small max sidelobe level.
Accuracy measure of single tone  wide main-lobe
Observed signal
Window wish list
NB: Strong DC component can shadow nearby small signals. Remove it!
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6. DFT - Window loss remedial
Smooth data-tapering windows cause information loss near edges.
Solution:
sliding (overlapping) DFTs.
Attenuated inputs get next window’s full gain & leakage reduced.
Usually 50% or 75% overlap (depends on main lobe width).
Drawback: increased total processing time.
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7. Zero padding
Improves DFT frequency inter-sampling spacing (“resolution”).
After padding frequencies NS = original samples, L = padded.
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8. DFT spectral resolution
Zero padding -2
DFT spectral resolution
Capability to distinguish two closely-spaced frequencies: not improved by zero-padding!.
Frequency inter-sampling spacing: increased by zero-padding (DFT “frequency span” unchanged due to same sampling frequency)
Additional reason for zero-padding: to reach a power-of-two input samples number (see FFT).
Zero-padding in frequency domain increases sampling rate in time domain. Note: it works only if sampling theorem satisfied!
Apply zero-padding after windowing (if any)! Otherwise stuffed zeros will partially distort window function.
NOTE
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9. DFT - scalloping loss (SL)
kmax
bin, k
We’re lucky here!
Input frequency f0 btwn. bin centres causes magnitude loss
SL = 20 Log10(|cr+kmax /ckmax|) ~
Worst case when f0 falls exactly midway between 2 successive bins (|r|=½) SL
bin, k
|r|  ½
f0 = (kmax + r) fS/N
Frequency error: f = r fS/N,
relative error: R=f / f0 = r/[(kmax+r)] R  1/(1+2 kmax)
kmax f0
Note: Non-rectangular windows broaden DFT main lobe  SL less severe
Correction depends on window used.
May impact on data interpretation (wrong f0!)
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10. DFT - SL Example SL remedial • increasing N (?) • improve windowing,
DC bias correction, Rectang. window, zero padding, FFT
• increasing N (?)
• improve windowing,
• zero-padding,
• interpolation around kmax.
SL remedial
DC bias correction, Hanning window, zero padding, FFT
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11. DFT - parabolic interpolation
Rectangular window
Hanning window
Parabolic interpolation often enough to find position of peak (i.e. frequency).
Other algorithms available depending on data.
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12. DFT averaging Incoherent averaging Coherent averaging
M = No. of DFT to average
k = bin number, k=0, 1 .. N-1
Background noise fluctuations reduced, average noise power unchanged.
Coherent averaging
Background noise power is reduced.
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13. Efficient DFT calculation: FFT
WNn,k = twiddle factors
k = 0,1 .. N-1
DFT
Direct DFT calculation redundancy
WNkn periodic function calculated many times.
Direct DFT calculation requires ~N2 complex multiplications.
complexity O(N2)
VERY BAD !
Algorithms (= Fast Fourier Transform) developed to compute N-points DFT with ~ Nlog2N multiplications (complexity O(Nlog2N) ).
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14. FFT advantages
DFT  N2
FFT  N log2N N
DFT
Radix-2 4 16 32
1024 80
128
16384
448
5120
DSPs & PLDs influenced algorithms design. ‘60s & ‘70s: multiplication counts was “quality factor”. Now: number of additions & memory access (s/w) and communication costs (h/w) also important.
NB: Usually you don’t want to write an FFT algorithm, just to “borrow” it !!! Go “shopping” onto the web!
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15. FFT philosophy
General philosophy (to be applied recursively): divide & conquer.
cost(sub-problems) + cost(mapping) < cost(original problem)
Different algorithms balance costs differently.
Example: Decimation-in-time algorithm
time frequency
Step 1: Time-domain decomposition. N-points signal  N, 1-point signals (interlace decomposition). Shuffled input data (bit-reversal). log2N stages.
(*): only first decomposition shown.
(*)
Step 2: 1-point input spectra calculation. (Nothing to do!)
Step 3: Frequency-domain synthesis N spectra synthesised into one.
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16. FFT family tree Divide & conquer N : GCD(*)(N1,N2) = 1
N1, N2 co-prime. Ex: 240 = 16·3·5
Cost: SUB-PROBLEMS.
No twiddle-factors calculations.
Easier mapping (permutations).
Some algorithms:
Good-Thomas, Kolba,
Parks, Winograd.
(*) GCD= Greatest Common Divisor
N : GCD(N1,N2) <> 1
Ex: N = 2n
Cost: MAPPING.
Twiddle-factors calculations.
Easier sub-problems.
Some algorithms:
Cooley-Tukey,
Decimation-in-time / in-frequency
Radix-2, Radix-4,
Split radix.
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17. (Some) FFT concepts & notes
Butterfly: basic FFT calculation element.
Decimation-in-time  time data shuffling.
Decimation-in-frequency  frequency data shuffling.
In-place computation: no auxiliary storage needed, allowed by most algorithms.
DFT pruning: only few bins needed or different from zero  only they get calculated (ex: Goertzel algorithm).
Real-data case: Mirroring effect in DFT coeffs.  only half of them calculated.
N power-of-two: Many common FFT algorithms work with power-of-two number of inputs. When they are not  pad inputs with zeroes.
Dual approach: data to be reordered in time or in frequency!
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18. Systems spectral analysis (hints)
System analysis: measure input-output relationship.
h[t] = impulse response
Linear Time Invariant
x[n] h[n]
y[n] predicted from { x[n], h[t] }
X(f) H(f) Y(f) = X(f) · H(f)
H(f) : LTI transfer function
Transfer function can be estimated by Y(f) / X(f)
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19. Estimating H(f) (hints)
Power Spectral Density of x[t] (FT of autocorrelation).
Cross Power Spectrum of x[t] & y[t] (FT of cross-correlation).
Transfer Function (ex: beam !)
Coherence function
values in [0,1]
assess degree of linear relationship between x[t] & y[t].
It is a check on H(f) validity!
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20. References - 1
Papers
Tom, Dick and Mary discover the DFT, J. R. Deller Jr, IEEE Signal Processing Magazine, pg , April 1994.
On the use of windows for harmonic analysis with the Discrete Fourier Transform, F. J. Harris, IEEE Proceedings, Vol. 66, No 1, January 1978.
Some windows with a very good sidelobe behaviour, A. H. Nuttall, IEEE Trans. on acoustics, speech and signal processing, Vol ASSP-29, no. 1, February 1981.
Some novel windows and a concise tutorial comparison of windows families, N. C. Geckinli, D. Yavuz, IEEE Trans. on acoustics, speech and signal processing, Vol ASSP-26, no. 6, December 1978.
Study of the accuracy and computation time requirements of a FFT-based measurement of the frequency, amplitude and phase of betatron oscillations in LEP, H.J. Schmickler, LEP/BI/Note
Causes et corrections des erreurs dans la mesure des caracteristiques des oscillations betatroniques obtenues a partir d’une transformation de Fourier, E. Asseo, CERN PS 85-9 (LEA).
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21. References - 2
Precise measurements of the betatron tune, R. Bartolini et al., Particle Accel., 1996, vol. 55, pp
How the FFT gained acceptance, J. W. Cooley, IEEE Signal Processing Magazine, January 1992.
A comparative analysis of FFT algorithms, A. Ganapathiraju et al., IEEE Trans.on Signal Processing, December 1997.
Books
The Fourier Transform and its applications, R. N. Bracewell, McGraw-Hill, 1986.
A History of scientific computing, edited by S. G. Nash, ACM Press, 1990.
Introduction to Fourier analysis, N. Morrison, John Wiley & Sons, 1994.
The DFT: An owner’s manual for the Discrete Fourier Transform, W. L. Briggs, SIAM, 1995.
The FFT: Fundamentals and concepts, R. W. Ramirez, Prentice Hall, 1985.
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